By Theorem REF , we may assume \(Y=\mathrm {Spa}(A)\)  for an affinoid \(K\) -algebra \(A\) . In particular, \(X\)  and \(Y\)  are both qcqs. We may clearly also assume that \(Y\)  is reduced. Our task is to show that \(Rf_*\)  preserves \(D^b_{zc}(-,\mathbf {Z}/n)\)  in this situation. As \(X\)  is quasi-compact, we may apply Proposition REF , so it suffices to show that \(Rf_* \mathbf {Z}/n \in D^b_{zc}\) . In fact, as the fibres of \(f\)  have bounded dimension (e.g., by checking on formal models), we know by cohomological dimension estimates and proper base change that \(Rf_*\)  has finite cohomological dimension, so it is enough to show that \(Rf_* \mathbf {Z}/n \in D_{zc}\) , i.e., that each \(R^i f_*\mathbf {Z}/n\)  is Zariski-constructible on \(Y\) . By proper base change and induction on \(\dim (Y)\) , it suffices to find a dense open \(U \subset Y\)  such that \((R^if_* \mathbf {Z}/n)|_U\)  is locally constant. As \(K\)  has characteristic 0, Temkin's [1]} gives a proper hypercover \(\epsilon :X^\bullet \rightarrow X\)  with each \(X^i\)  being \(K\) -smooth. Cohomological descent enables us to compute \(Rf_* \mathbf {Z}/n\)  as the totalization of \(Rg_* \mathbf {Z}/n\) , where \(g = f \circ \epsilon \) . As \(\mathcal {H}^i\)  of the totalization of a cosimplicial object \(K(\bullet ):\Delta \rightarrow \mathcal {D}^{\ge 0}\)  only depends on truncated cosimplicial object \(K(\bullet )|_{\Delta _{\le i+1}}\) , we can replace \(X^\bullet \)  with the finite diagram \(X^{\le i+1}\)  and then by each \(X^j\)  to assume that \(X\)  is smooth. As \(Y\)  is reduced, our generic smoothness result (Theorem REF ) yields a Zariski-dense Zariski-open \(U \subset X\)  such that \(f:X \rightarrow Y\)  is smooth over \(U\) . It then suffices to show that \(R^i g_* \mathbf {Z}/n\)  is locally constant when \(g\)  is both proper and smooth. To check this, we may assume \(n=\ell \)  is a prime. Now if \(\ell \ne p\) , the claim reduces to [2]}, while the claim for \(\ell = p\)  reduces to [3]}.
